Technically, 'anime' just means animation, of any kind, period. It's just 'animation' shortened to 3 syllables instead of the 6 needed to say it in Japanese.
While it might've been so when the word was borrowed from english, it's certainly not a case now. Anime, as a style of Japanese animation, has evolved in such ways as to become separate, and easily distinguishable by all, subset of the main body of animation.
Today, it's incorrect to call e.g. Toystory an anime, while of course, it's still true to call e.g.Akira an animation.
It's like saying that Bollywood is a medium. Anime is an industry, animation is a medium
A medium is any method or mode of transmitting information. Bollywood movies would count as a medium.
Anime is ANY animation from Japan and people often expand that definition to include any animation that uses that style. At the very least it is considered a "Style" (thus subset) of Animation. Though I am open to being incorrect.
"Style" and "industry" are corelated in such a close fashion, that I'm very well inclined to treat these as the same. Anime industry caters to a certain, unchanged for decades, audience, producing output of certain style, which pleases(is bought by) that very group.
I do not agree with calling anime or Bollywood a medium. Medium, as you said, is a mode of transmitting information, it's not information by itself. The two examples mentioned above contain within their definition a set of rules as to
what stories can be told by them, not just
how.